Finite Group Databasebeta

← #22592652

Subgroups — #22592652

Order 9088.

584 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556090C211yesmaximal, characteristic
2C2×C4⋊C4×D71C211yesmaximal, characteristic, complemented
3#17985938C211yesmaximal, characteristic, complemented
4#17985936C211yesmaximal, characteristic, complemented
5#17985934C211yesmaximal, characteristic, complemented
6#17985896C212yesmaximal, complemented
7#17985877C212yesmaximal
8#17985873C212yesmaximal, complemented
9#17985858C212yesmaximal, complemented
10#17985858C211yesmaximal, characteristic, complemented
11#400×C71C211yesmaximal, characteristic, complemented
12C22×D284C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15C22×C4×D71C2212yes—
16#14556090—24no—
17C2×C4⋊Dic71C2211yescharacteristic
18C2×C4⋊Dic71C2212yes—
19C2×Dic71⋊C4C2212yes—
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2212yes—
22C4⋊C4×D71—24no—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×C4×Dic71C2211yescharacteristic
29C71⋊#95C2212yes—
30C71⋊#95C2211yescharacteristic
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34#95×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C23×D71C2311yescharacteristic
39C2×D284—22no—
40C2×D284—22no—
41C2×D284Q814yes—
42C22×Dic71C2312yes—
43C22×Dic71C2312yes—
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2311yescharacteristic
46C2×C4×D71—24no—
47C2×C4×D71—24no—
48C2×C4×D71—22no—
49C2×C4×D71—24no—
50C2×C4×D71—42no—
51C2×C4×D71—42no—
52C2×C4×D71D414yes—
53C4⋊Dic71—22no—
54C4⋊Dic71—42no—
55Dic71⋊C4—24no—
56Dic71⋊C4—24no—
57Dic71⋊C4—42no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—42no—
62(C2×D71)⋊C4—42no—
63C4×Dic71—22no—
64C22×C4×C71C2311yescharacteristic
65C22×C4×C71C2312yes—
66C22×C4×C71C2312yes—
67C22×C4×C71C2311yescharacteristic
68C22×C4×C71C2311yescharacteristic
69C4⋊C4×C71—22no—
70C4⋊C4×C71—22no—
71C23×C71C2411yescharacteristic
72C22×D71—22no—
73C22×D71—22no—
74C22×D71—21no—
75C22×D71—22no—
76C22×D71—21no—
77C22×D71—22no—
78C22×D71C2×D412yes—
79C22×D71C2×Q812yes—
80C22×D71C2×Q812yes—
81C22×D71#4112yes—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71C2×D412yes—
94C2×C4×C71C2×Q812yes—
95C2×Dic71—21no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71—22no—
103C2×Dic71C2×D412yes—
104C2×Dic71#4114yes—
105D284—28no—
106C4×D71—28no—
107C4×D71—48no—
108C4×D71—44no—
109C22×C71C22×D411yescharacteristic
110C22×C71C22×Q811yescharacteristic
111C22×C71C2×#4111yescharacteristic
112C22×C71C2×#4112yes—
113C22×C71H(5,𝔽2)12yes—
114C2×D71—22no—
115C2×D71—22no—
116C2×D71—24no—
117C2×D71—24no—
118C2×D71—22no—
119C2×D71—24no—
120C2×D71—41no—
121C2×D71—41no—
122C2×D71—42no—
123C2×D71C22⋊Q814yes—
124C4×C71—22no—
125C4×C71—22no—
126C4×C71—41no—
127C4×C71—42no—
128C4×C71—42no—
129Dic71—42no—
130Dic71—41no—
131Dic71—42no—
132Dic71#12314yes—
133C2×C71C2×C22⋊Q811yescharacteristic
134C2×C71C2×#12311yescharacteristic
135C2×C71#53511yescharacteristic
136C2×C71#55312yes—
137C2×C71#55412yes—
138D71—24no—
139D71—42no—
140#2487—711nomaximal, Sylow
141C71#248711yescharacteristic, complemented, Sylow
142C2×C4×D4—711no—
143C22×C4⋊C4—711no—
144#400—711no—
145#400D7111yescharacteristic, complemented
146#399—711no—
147#399—711no—
148#389—712no—
149#386—712no—
150#385—712no—
151#380—712no—
152#380—711no—
153C22×D4—711no—
154C23×C4—712no—
155C23×C4—711no—
156C23×C4—711no—
157C4×D4—1424no—
158C2×C4⋊C4—712no—
159C2×C4⋊C4—712no—
160C2×C4⋊C4—712no—
161C2×C4⋊C4—711no—
162C2×C4⋊C4—712no—
163C2×C4⋊C4—1424no—
164C2×C4⋊C4C2×D7111yescharacteristic
165C2×C4⋊C4C2×D7111yescharacteristic
166C2×C22⋊C4—712no—
167C2×C22⋊C4—712no—
168C2×C22⋊C4—711no—
169C2×C22⋊C4—711no—
170C2×C22⋊C4—712no—
171C2×C42—711no—
172#95—712no—
173#95—711no—
174#95—712no—
175#95C2×D7112yes—
176#95C2×D7111yescharacteristic
177#95C2×D7112yes—
178C24—711no—
179C24—711no—
180C2×D4—714no—
181C2×D4—1422no—
182C2×D4—1422no—
183C22×C4—2842no—
184C22×C4—2842no—
185C22×C4—711no—
186C22×C4—714no—
187C22×C4—712no—
188C22×C4—712no—
189C22×C4—711no—
190C22×C4—1422no—
191C22×C4—1424no—
192C22×C4—1424no—
193C22×C4—1424no—
194C22×C4C22×D7111yescharacteristic
195C22×C4C22×D7112yes—
196C22×C4C22×D7112yes—
197C22×C4C22×D7111yescharacteristic
198C22×C4C22×D7111yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—2842no—
202C4⋊C4—2842no—
203C4⋊C4—1424no—
204C4⋊C4—1424no—
205C4⋊C4—1422no—
206C22⋊C4—2842no—
207C22⋊C4—2842no—
208C22⋊C4—1424no—
209C22⋊C4—1422no—
210C22⋊C4—1422no—
211C42—1422no—
212C23—712no—
213C23—712no—
214C23—712no—
215C23—712no—
216C23—1421no—
217C23—1422no—
218C23—1421no—
219C23—1422no—
220C23—1422no—
221C23—1422no—
222C23C23×D7111yescharacteristic
223D4—1428no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—2848no—
236C2×C4—2844no—
237C2×C4—712no—
238C2×C4—714no—
239C2×C4—1428no—
240C2×C4—1421no—
241C2×C4—1422no—
242C2×C4—1422no—
243C2×C4—1422no—
244C2×C4—1422no—
245C2×C4—1422no—
246C2×C4—1422no—
247C2×C4—1422no—
248C2×C4D4×D7112yes—
249C2×C4Q8×D7112yes—
250C22—2841no—
251C22—2842no—
252C22—2841no—
253C22—714no—
254C22—1424no—
255C22—1422no—
256C22—1422no—
257C22—1424no—
258C22—1424no—
259C22—1422no—
260C22C2×D4×D7111yescharacteristic
261C22C2×Q8×D7111yescharacteristic
262C22#41×D7111yescharacteristic
263C22#41×D7112yes—
264C22#1455617412yes—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—42no—
269C4—41no—
270C4—2842no—
271C4—2841no—
272C4—2842no—
273C4—714no—
274C2—2842no—
275C2—1424no—
276C2C22⋊Q8×D7111yescharacteristic
277C2#123×D7111yescharacteristic
278C2#1798656711yescharacteristic
279C2#1798670012yes—
280C2#1798670512yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2487
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4⋊C4×D71C2semidirect
3. #17985938C2semidirect
4. #17985936C2semidirect
5. #17985934C2semidirect
6. #17985896C2semidirect
8. #17985873C2semidirect
9. #17985858C2semidirect
10. #17985858C2semidirect
11. #400×C71C2semidirect
141. C71#2487semidirect
145. #400D71semidirect